Points $A$, $B$, $Q$, $D$, and $C$ lie on the circle shown and the measures of arcs $BQ$ and $QD$ are $42^\circ$ and $38^\circ$, respectively.  Find the sum of the measures of angles $P$ and $Q$, in degrees.

[asy]
import graph;

unitsize(2 cm);

pair A, B, C, D, P, Q;

A = dir(160);
B = dir(45);
C = dir(190);
D = dir(-30);
P = extension(A,B,C,D);
Q = dir(0);

draw(Circle((0,0),1));
draw(B--P--D);
draw(A--Q--C);

label("$A$", A, NW);
label("$B$", B, NE);
label("$C$", C, SW);
label("$D$", D, SE);
label("$P$", P, W);
label("$Q$", Q, E);
[/asy]
Answer: We have that $\angle P = (\text{arc } BD - \text{arc } AC)/2$ and $\angle Q = (\text{arc } AC)/2$.  Hence, $\angle P + \angle Q = (\text{arc } BD)/2 = (42^\circ + 38^\circ)/2 = \boxed{40^\circ}$.